HAUSDORFF DERIVATIVE MODEL FOR CHARACTERIZATION OF NON-FICKIAN MIXING IN FRACTAL POROUS MEDIA

Abstract

This study investigates the scalar dissipation rate (SDR) and dilution index of non-Fickian mixing by using the Hausdorff derivative model for conservative and first-order decaying tracers under different boundary conditions. The expressions of SDR and dilution index are derived based on the analytical solution of the Hausdorff derivative model, in which the time and space Hausdorff derivative orders, respectively capture the complexity in transport trajectory and transport scale. The properties of SDR and dilution index are discussed for different cases of Peclet number, decaying rate of the radioactive tracer, and Hausdorff derivative order, respectively. We find that the SDR of non-Fickian mixing decays more slowly than that of the Fickian diffusion, and the time scale deviates from [Formula: see text]. The evolution of the SDR has a sharp peak and decays very fast to zero when Peclet number is large. For the radioactive tracer, the larger values of decay rate, the smaller values of SDR, which decays faster to zero. The Hausdorff derivative model with larger Peclet number leads to larger dilution index. The dilution index is larger for smaller decay rate before reaching the equilibrium state. Consequently, the two metrics can be satisfactorily used to describe non-Fickian mixing based on the Hausdorff derivative model. Future studies should be designed to examine the evolution of SDR and dilution index in real geological and hydrological systems undergoing structure changes and chemical reactions.